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 In 1854, nearly twenty years later, Gauss heard from his pupil, Riemann, a marvellous dissertation carrying the discussion one step further by developing the notion of $\scriptstyle{n}$-ply extended magnitude, and the measure-relations of which a manifoldness of $$\scriptstyle{n}$$ dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen Before this the idea of $$\scriptstyle{n}$$ dimensions had suggested itself under various aspects to Lagrange, Plücker, and H. Grassmann. About the same time with Riemann's paper, others were published from the pens of Helmholtz and Beltrami. These contributed powerfully to the victory of logic over excessive empiricism. This period marks the beginning of lively discussions upon this subject. Some writers—Bellavitis, for example—were able to see in non-Euclidean geometry and $\scriptstyle{n}$-dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularised the subject in lectures, and in articles for various magazines.

Eugenio Beltrami, born at Cremona, Italy, in 1835, and now professor at Rome, wrote the classical paper Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6), which is analytical (and, like several other papers, should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant